LOGOS Framework v9: Unified Variational Theory of Cyclic Delay Networks

Version 9 of the LOGOS framework consolidates and closes the coretheoretical programme developed over versions v1 through v8. Priorwork established, analytically, the Hopf bifurcation structure ofcyclic delay-differential networks — universal frequencyomegac = sqrt (g² - 1), delay-sum invariance, and the supercriticalfirst Lyapunov coefficient l1 ≈ -3. 47 < 0 — and, heuristically, acollection of modular operational constructs (prediction error, memorykernel, chaos budget, adaptive control interval, temporal warping, noise filtering, attractor dominance). The present paper unifies these two tracks: Part I introduces a single action functionalJ = integral of TW (t) * L (S) - alpha*E² - beta*Rchaos dtfrom which all five operational modules are derived as necessaryvariational conditions. No module remains heuristic. Part II derives the stochastic Hamilton-Jacobi-Bellman (HJB) equationfor J under the centre-manifold amplitude dynamics of the cyclic delaynetwork, and proves the main result: V₄ = -l₁ / (2*omegac) ≈ +0. 448 * TWbar The first Lyapunov coefficient l₁ (computed analytically in v3) isexactly the quartic curvature coefficient of the optimal valuefunction at the Hopf bifurcation point. Three consequences follow: (i) l₁ is identified as the curvature of the optimal value function; (ii) the supercritical limit cycle is the unique global maximiser of Jin the post-bifurcation regime; (iii) the Temporal Warping factorTW (t) enters the HJB equation as an effective discount rate, establishing a formal equivalence between cooperative coherence andtemporal depth of optimal planning. With this consolidation, the core mathematical programme of LOGOS iscomplete. Every previously independent result — from delay-suminvariance to the variational derivation of the noise filter — nowhas a formal place within a single variational structure. A closing section collects three precisely-stated open problems leftdeliberately for future work: (1) a Cₜemp metric for temporalbandwidth and memory depth, (2) Floquet stability analysis of thesupercritical limit cycle, and (3) N-layer generalisation beyond thecurrent N=3 proofs. Each can be pursued incrementally withoutreopening the core framework. Numerical verification: DDE core (N=3, g=4, tau₁=2. 0, r=phi) isintegrated with the variational layer. 31 attractor transitions inT=1500 time units under the optimal noise structure, with Bchaosdipping to 0. 839 at switch events — consistent with the Kramersescape rate predicted by the HJB value function.